Approximate amenability and contractibility of hypergroup algebras

Abstract

Let $K$ be a hypergroup. The purpose of this article is to study the notions of amenability of the hypergroup algebras $L\left(K\right)$, $M\left(K\right)$, and $L(K{)}^{\ast \ast }$. Among other results, we obtain a characterization of approximate amenability of $L(K{)}^{\ast \ast }$. Moreover, we introduce the Banach space $L_{\infty }(K,L(K\left)\right)$ and prove that the dual of a Banach hypergroup algebra $L\left(K\right)$ can be identified with $L_{\infty }(K,L(K\left)\right)$. In particular, $L\left(K\right)$ is an $F$-algebra. By using this fact, we give necessary and sufficient conditions for $K$ to be left-amenable.